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Properties

Label 2-1710-95.49-c1-0-21
Degree $2$
Conductor $1710$
Sign $0.624 + 0.781i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.919 − 2.03i)5-s + 1.07i·7-s − 0.999i·8-s + (−0.223 + 2.22i)10-s − 0.410·11-s + (−3.30 + 1.91i)13-s + (0.537 − 0.931i)14-s + (−0.5 + 0.866i)16-s + (1.08 + 0.627i)17-s + (3.85 + 2.03i)19-s + (1.30 − 1.81i)20-s + (0.355 + 0.205i)22-s + (3.23 − 1.86i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.410 − 0.911i)5-s + 0.406i·7-s − 0.353i·8-s + (−0.0706 + 0.703i)10-s − 0.123·11-s + (−0.917 + 0.529i)13-s + (0.143 − 0.248i)14-s + (−0.125 + 0.216i)16-s + (0.263 + 0.152i)17-s + (0.884 + 0.466i)19-s + (0.291 − 0.405i)20-s + (0.0758 + 0.0437i)22-s + (0.674 − 0.389i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.624 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.052954021\)
\(L(\frac12)\) \(\approx\) \(1.052954021\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.919 + 2.03i)T \)
19 \( 1 + (-3.85 - 2.03i)T \)
good7 \( 1 - 1.07iT - 7T^{2} \)
11 \( 1 + 0.410T + 11T^{2} \)
13 \( 1 + (3.30 - 1.91i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.08 - 0.627i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.23 + 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.18 + 2.05i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + 2.08iT - 37T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.92 - 1.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.58 - 2.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.32 + 2.49i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.25 - 2.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.37 + 5.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.07 + 4.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.79 - 8.30i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.02 + 3.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.47 + 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.12iT - 83T^{2} \)
89 \( 1 + (-2.72 - 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.5 - 6.08i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.255203294270053757488771277328, −8.526303206399660240886351636795, −7.79006270359780795190070248521, −7.12902767957916820515620043076, −5.95172648759059984177222952677, −5.02216373462855210425972537054, −4.21605897117625013680910342432, −3.07168530394494723123333613454, −1.96775897276937403729343375485, −0.70242771721269930010662762432, 0.860076557159579192959468012616, 2.53729076708864159163825586652, 3.29252351402431800654679783911, 4.58044580630981911594557921571, 5.48583111977840603322671561978, 6.50074113604867797119192092951, 7.30838082568864974511676622476, 7.61502481572123268816574012484, 8.547697962996287313862229719000, 9.588405667335765060994937129175

Graph of the $Z$-function along the critical line